Application of Meshless Methods for Thermal Analysis

Many numerical and analytical schemes exist for solving heat transfer problems. The meshless method is a particularly attractive method that is receiving attention in the engineering and scientific modeling communities. The meshless method is simple, accurate, and requires no polygonalisation. In this study, we focus on the application of meshless methods using radial basis functions (RBFs) — which are simple to implement — for thermal problems. Radial basis functions are the natural generalization of univariate polynomial splines to a multivariate setting that work for arbitrary geometry with high dimensions. RBF functions depend only on the distance from some center point. Using distance functions, RBFs can be easily implemented to model heat transfer in arbitrary dimension or symmetry

Application of local RBF collocation method to prediction of mechanics-related phenomena during dc casting of aluminium alloys

A meshless local collocation method using radial basis functions (LRBFCM) has been developed to model the thermomechanical phenomena during the process of DC casting. The model uses elastic-viscoplastic constitutive relations to describe the inhomogeneous material below the coherency isotherm. It is coupled with the results of the fluid flow model, which is used to determine the computational domain and to calculate the thermal strain

A RBF-Based local collocation method for modelling thermomechanical phenomena during DC casting of aluminium billets

In this work, the local radial basis function collocation method is applied to the thermoelasticity with intention to model the low-frequency electromagnetic direct-chill casting process of aluminium billets. The devised thermoelastic model is coupled with the heat transport model for the DC casting process and preliminary results on the stress state are presented. The effect of the casting speed and the application of the electromagnetic field on the principal stresses is presented

Simulation of macrosegregation benchmark on a non-uniform computational node arrangement with a meshless method

An application of a meshless numerical method on a macrosegregation benchmark case is developed in the present paper. The test case is solidification in 2D rectangular cavity, filled with liquid metal and chilled from both sides. This is a highly non-linear problem due to a strong coupling of the macroscopic transport equations with the microsegregation model. The main result is the macrosegregation pattern of the solidified metal Al4.5wt%Cu alloy is used for evaluation of the problem. The model uses diffuse approximate meshless method with the second-order polynomial basis for spatial integration and explicit time-stepping. Simulations are performed on uniform and non-uniform computational node arrangements and compared to each other. The results on uniform and non-uniform node arrangements show a very good matching with the finite volume method results and results based on radial basis function collocation method. This shows that diffuse approximate method based on non-uniform node arrangements can be used for solving macrosegregation problems

Simulation of macrosegregation in low-frequency electromagnetic casting by a meshless method

The novel use of a meshless numerical approach for simulation of macrosegregation in the low-frequency electromagnetic casting is presented along with the analysis of the simulation results. The casting model includes a coupled set of mass, momentum, energy, and species conservation equations. Lorentz force is computed with the induction equation and used in the solidification model. The coupled physical model is solved in cylindrical coordinate system and can be used to model aluminium alloy billet production. Explicit scheme is used for the temporal discretization, while the meshless diffuse approximate method is used for the spatial discretization. The method is localized with subdomains containing 14 local nodes. The Gaussian weight is used in the weighted least squares minimization. Furthermore, the Gaussian is shifted upstream, when an upwind effect is required in order to increase the convection stability. Direct chill casting under the influence of electromagnetic field (EMF) is simulated for various electric amplitudes and currents. The material properties of Al-5.25wt%Cu are used. The casting parameters and material properties are constant in all presented simulations, while EMF is turned off in some cases in order to study its effect on solidification. The results show that EMF has a large effect on the melt-flow and solidification. Oscillatory, instead of a steadystate, solution is obtained in case of certain geometries in EMF casting. The effect of EMF is hard to predict without the use of numerical simulations, due to strong coupled effects of casting geometry, casting parameters, and EMF parameters. This shows the need for numerical modelling of this strongly coupled problem for its better understanding

Computational modelling of gas focused thin liquid sheets

Formation of liquid sheets has been demonstrated as a critical capability needed in many different research fields. Many different types of liquid sheets have been produced experimentally, its thickness ranging from few tens of nanometres to few micrometres. Due to the small size of such systems, where physical parameters such as thickness, velocity and temperature are difficult to measure, a need for numerical simulation of liquid sheets arises. In this paper we demonstrate such capability with sheets that can be used in experiments with synchrotrons, X-ray free electron lasers or lab sources. A modified gas dynamic virtual nozzle (GDVN) design is used in order to generate micrometre thin sheets. The system is characterised by a strongly coupled problem between the focusing gas flow and the liquid sheet flow. Investigation of varying physical properties of liquid is performed in order to demonstrate the effects on the sheet production. It was found that the primary sheet thickness is not sensitive to the variation of liquid viscosity and density. On the other hand, the variation of surface tension greatly affects the thickness and the width of a primary sheet, such as expected in flows where surface tension is the dominating force. Findings demonstrate that by lowering the surface tension of a liquid, i.e. changing liquid from water to alcohol for example, would produce thinner and wider sheets. Simulations were produced with OpenFOAM, relying on finite volume based multiphase solver “compressibleInterFoam”, capable of simulating free surfaces. Mixture formulation of a multiphase system consists of an incompressible liquid phase along with a compressible ideal gaseous phase. Such model was also used in axisymmetric GDVN micro-jet simulations preformed in our previous work. Due to the need for 3D simulations and huge computational resources needed, an adaptive approach was chosen. This made the simulations of liquid sheets of thicknesses down to 500 nm possible

Modelling of electromagnetic breaking and electromagnetic stirring in the process of continuous casting of steel

More than 95 % of crude steel is nowadays processed by Continuous Casting (CC) [1]. To further advance the quality of the products and efficiency of the process, electromagnetic (EM) field, which affects the fluid flow as well as the temperature and segregation is added to the CC process. In general, there are two types of electromagnetic devices applicable to the CC process; the electromagnetic breakers (EMBR) which employ the direct current, and the electromagnetic stirrers (EMS), which employ the alternating current. Which of the devices is employed depends on what are the desired effects. Both of the processes are modelled by implementing the Lorentz force into the momentum equation, and if necessary, the Joule heating term into the energy equation. However, the way how these two terms are modelled, depends on the type of the implemented device. In case of EMBRs, the assumption of low magnetic Reynolds number Rem is made, and consequently, the current density is calculated by solving the Poisson’s equation for the electric potential. The EMSs on the other hand, require a low-frequency approximation and the solution of induction equation. The complete set of governing equations for CC process [2] under the influence of magnetic field includes mass, momentum, energy, and species transfer equations, and Maxwell’s equations together with Ohm’s law and charge conservation equation. Additionally, the turbulent kinetic energy and dissipation rate equations together with Abe-Kondoh-Nagano closures are used to account for the turbulence, the lever rule model is used to model the microsegregation, the mixture continuum model is used to model the macrosegregation, fractional step method is used to model pressure-velocity coupling and the enthalpy-temperature relation is used to calculate the temperature from the enthalpy. The solution is sought for on a five-nodded local subdomains by constructing an approximation with multiquadric radial basis functions as a basis and collocation to find the expansion coefficients [3,4]. Present paper presents the discretization of governing equations, together with boundary conditions for both EMBR and EMS devices with meshless Local Radial Basis Function Collocation Method (LRBFCM) [5]

Meshless modelling of microstructure evolution in the continuous casting of steel

A two-dimensional two-scale slice model has been developed to predict the microstructure evolution in the solidifying strand with an arbitrary cross section geometry during continuous casting of steel. The enthalpy equation is solved at the macro level by using meshless local radial basis function collocation method (LRBFCM) based on multiquadrics for spatial discretization and explicit Euler scheme for temporal discretization. The temperature and the solid fraction in computational nodes are calculated by using a continuum model formulation while the lever rule is used as the supplementary microsegregation relation. The temperature field is interpolated to the micro level by using LRBFCM. At the micro level, the normal distribution and Kurz-Giovanola-Trivedi model are proposed to determine temperature dependent nucleation rate and grain growth velocity, respectively. Meshless point-automata algorithm is applied to implement nucleation and grain growth equations. Several examples of computations of the strand with different cross-sections are shown

Numerical simulations of micro jets produced with a double flow focusing nozzle

Stable and reliable micro jets are important for many applications. Double flow focused micro jets are a novelty with an important advantage of significantly reduced sample consumption. Numerical simulations of double flow focused micro jets are a highly complex task. They represents a great computational challenge due to the multiphase nature of the problem, strong coupling between the gas and the two liquids and the sub-micron size cells needed. Simulations were performed with the open source computational fluid dynamics toolbox called OpenFOAM. Two multiphase solvers were used, one of which was modified in order to properly describe the interface between the focusing liquid and the gas. In this study two different incompressible physical models were considered and compared. A model with no mixing of the two fluids (multiphaseInterFoam solver) and a model where the diffusion of the two fluids is permitted (modified interMixingFoam solver). The results of simulations for the two different physical models using the same inlet parameters are presented. Additionally, a parametric analysis for the mixing case was performed to study the effects of different parameters on the jet formation. Particularly how the different diffusion values couple with the jet length, diameter and its stability. Results show a match in jet diameter and jet length for both models when the same set of parameters is used

A RBF-Based local collocation method for modelling thermomechanical phenomena during DC casting of aluminium billets

In this work, the local radial basis function collocation method is applied to the thermoelasticity with intention to model the low-frequency electromagnetic direct-chill casting process of aluminium billets. The devised thermoelastic model is coupled with the heat transport model for the DC casting process and preliminary results on the stress state are presented. The effect of the casting speed and the application of the electromagnetic field on the principal stresses is presented